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Neuronal Dynamics: Computational Neuroscience of Single Neurons Week 2 – Biophysical modeling: The Hodgkin-Huxley model
Wulfram Gerstner EPFL, Lausanne, Switzerland
2.1 Biophysics of neurons - Overview 2.2 Reversal potential
- Nernst equation 2.3 Hodgin-Huxley Model 2.4 Threshold in the Hodgkin-Huxley Model - where is the firing threshold? 2.5. Detailed biophysical models - the zoo of ion channels
Week 2 – part 5: Detailed Biophysical Models
inside
outside
Ka
Na
Ion channels Ion pump
Neuronal Dynamics – 2.5 Biophysical models
β α
Na+ channel from rat heart (Patlak and Ortiz 1985) A traces from a patch containing several channels. Bottom: average gives current time course. B. Opening times of single channel events
Steps: Different number of channels Ca2+
Na+
K+
Ions/proteins
Neuronal Dynamics – 2.5 Ion channels
inside
outside
Ka
Na
Ion channels Ion pump
Neuronal Dynamics – 2.5 Biophysical models
β α
There are about 200 identified ion channels
http://channelpedia.epfl.ch/
How can we know which ones are present in a given neuron?
β α
α
β
Kvβ1 Kvβ2 Kvβ3
Kvβ
Voltage Activated K+ Channels!
+ + Kv1.1 Kv1.2 Kv1.3 Kv1.4 Kv1.5 Kv1.6 Kv2.2 Kv2.1 Kv3.2
Kv3.1 Kv3.3 Kv3.4 KChIP1 KChIP2 KChIP3
KChIP
Kv1 Kv3 Kv2 Kv4 Kv5 Kv6 Kv8 Kv9
Kv
Kv4.3 Kv4.1 Kv4.2
Ion Channels investigated in the study of!
CGRP!
CB!
PV!
CR!
NPY!
VIP!
SOM!
CCK!
pENK
!Dy
n!
SP!
CRH!
schematic mRNA!Expression profile!
C gl gKv1 gNa
I
gKv3
Toledo-Rodriguez, …, Markram (2004)
C gl gKv1 gNa
I
gKv3
Model of a hypothetical neuron
)()()()()( 233
411
3 tIEugEungEungEuhmgdtduC llKKvKvKKvKvNaNa +−−−−−−−−=
)()(0
uumm
dtdm
mτ−
−=1 0,11
,1
( )( )n
n n udndt uτ
−= −
)()(0
uuhh
dtdh
hτ−
−=
stimulus NaI KI leakI
inside
outside
Ka
Na
Ion channels Ion pump
Erisir et al, 1999 Hodgkin and Huxley, 1952
1 0,33
,3
( )( )n
n n udndt uτ
−= −
How many parameters per channel?
C gl gKv1 gNa
I
gKv3
Model of a hypothetical neuron
I(t)
Spike
Subthreshold
Detailed model, based on ion channels
C gl gKv1 gNa
I
gKv3
Model of a hypothetical neuron (type I)
Biophysical model, based on ion channels
Current pulse
constant current
- Delayed AP initiation - Smooth f-I curve type I neuron
Neuronal Dynamics – 2.5 Adaptation
Functional roles of channels? - Example: adaptation
I
u
Neuronal Dynamics – 2.5 Adaptation: IM -current M current: - Potassium current - Kv7 subunits - slow time constant
IM current is one of many potential sources of adaptation Yamada et al., 1989
( )M M KI g m u E= −
Neuronal Dynamics – 2.5 Adaptation – INaP current current: - persistent sodium current - fast activation time constant - slow inactivation ( ~ 1s)
INaP current - increases firing threshold - source of adaptation Aracri et al., 2006
( )NaP NaP NaI g m h u E= −
inside
outside
Ka
Na
Ion channels Ion pump
Neuronal Dynamics – 2.5 Biophysical models
β
Hodgkin-Huxley model provides flexible framework
Hodgkin&Huxley (1952) Nobel Prize 1963
Exercise – 2. 5. Hodgkin-Huxley model – gating dynamics
( ) (1 ) ( )m mdm u m u mdt
α β= − −
0 ( ) 0.5{1 tanh[ ( )]m u uγ θ= + −
1( ) ( )1 exp[ ( ) / ]m mu u
u a bα β= =
− − +γ θ
A) Often the gating dynamics is formulated as
B) Assume a form
How are a and b related to and in the equations 0 ( )( )m
m m udmdt uτ
−= −
C ) What is the time constant ? ( )m uτ
0 ( )( )m
m m udmdt uτ
−= −
0 ( ) ( )mm u and uτCalculate
Neuronal Dynamics – References and Suggested Reading
Reading: W. Gerstner, W.M. Kistler, R. Naud and L. Paninski, Neuronal Dynamics: from single neurons to networks and models of cognition. Chapter 2: The Hodgkin-Huxley Model, Cambridge Univ. Press, 2014 OR W. Gerstner and W. M. Kistler, Spiking Neuron Models, Chapter 2, Cambridge, 2002
- Hodgkin, A. L. and Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol, 117(4):500-544. - Ranjan, R.,et al. (2011). Channelpedia: an integrative and interactive database for ion channels. Front Neuroinform, 5:36. - Toledo-Rodriguez, M., Blumenfeld, B., Wu, C., Luo, J., Attali, B., Goodman, P., and Markram, H. (2004). Correlation maps allow neuronal electrical properties to be predicted from single-cell gene expression profiles in rat neocortex. Cerebral Cortex, 14:1310-1327. - Yamada, W. M., Koch, C., and Adams, P. R. (1989). Multiple channels and calcium dynamics. In Koch, C. and Segev, I., editors, Methods in neuronal modeling, MIT Press. - Aracri, P., et al. (2006). Layer-specic properties of the persistent sodium current in sensorimotor cortex. Journal of Neurophysiol., 95(6):3460-3468.